Three-player impartial games

Three-player impartial games James Propp Department of Mathematics, University of Wisconsin (November 10, 1998) Past efforts to classify impartial three-player combinatorial games (the theories of Li [3] and Straffin [4]) have made various restrictive assumptions about the rationality of one s opponents and the formation and behavior of coalitions One may instead adopt an agnostic attitude towards such issues, and seek only to understand in what circumstances one player has a winning strategy against the combined forces of the other two By limiting ourselves to this more modest theoretical objective, and by regarding two games as being equivalent if they are interchangeable in all disjunctive sums as far as single-player winnability is concerned, we can obtain an interesting analogue of Grundy values for three-player impartial games 0 INTRODUCTION Let us begin with a very specific problem: Assume is an impartial (positional) game played by three people who alternate moves in cyclic fashion (Natalie, Oliver, Percival, Natalie, Oliver, Percival, ), under the convention that the player who makes the last move wins Let be another such game Suppose that the second player, Oliver, has a winning strategy for Suppose also that Oliver has a winning strategy for Is it possible for Oliver to have a winning strategy for the disjunctive sum as well? Recall that an impartial positional game is specified by (i) an initial position, (ii) the set of all positions that can arise during play, and (iii) the set of all legal moves from one position to another The winner is the last player to make a move To avoid the possibility of a game going on forever, we require that from no position may there be an infinite chain of legal moves The disjunctive sum of two such games is the game in which a legal move consists of making a move in (leaving alone) or making a move in (leaving alone) Readers unfamiliar with the theory of two-player impartial games should consult [1] or [2] It is important to notice that in a three-player game, it is possible that none of the players has a winning strategy The simplest example is the Nim game that starts from the position, where 1 and 2 denote Nim-heaps of size one and two respectively As usual, a legal move consists of taking a number of counters from a single heap In this example, the first player has no winning move, but his actions determine whether the second or third player will win the game None of the players has a winning strategy That is, any two players can cooperate to prevent the remaining player from winning It is in a player s interest to join such a coalition of size two if he can count on his partner to share the prize with him unless the third player counters by offering an even bigger share of the prize This kind of situation is well known in the theory of economic (as opposed to positional) games In such games, however, play is usually simultaneous rather than sequential Bob Li [3] has worked out a theory of multi-player positional games by decreeing that a player s winnings depend on how recently he has moved when the game ends (the last player to move wins the most, the player who moved before him wins the next most, and so on), and by assuming that each player will play rationally so as to get the highest winnings possible Li s theory, when applied to games like Nim, leads to quite pretty results, and this is perhaps sufficient justification for it; but it is worth pointing out that, to the extent that game theory is supposed to be applicable to the actual playing of games, it is a bit odd to assume that one s adversaries are going to play perfectly Indeed, the only kind of adversaries a sensible person would play with, at least when money is involved, are those who do not know the winning strategy Only in the case of two-player games is it the case that a player has a winning strategy against an arbitrary adversary if and only if he has a winning strategy against a perfectly rational adversary Phil Straffin [4] has his own approach to three-player games He adopts a policy ( McCarthy s revenge rule ) governing how a player should act in a situation where he himself cannot win but where he can choose which of his opponents will win Straffin analyzes Nim under such a revenge rule, and his results are satisfying if taken on their own terms, but the approach is open to the same practical objections as Li s Specifically, if a player s winning strategy depends on the assumption that his adversaries will be able to recognize when they can t win, then the player s strategy is guaranteed to work only when his opponents can see all the way to the leaves of the game tree In this case, at least one of them (and perhaps each of them) believes he can t win; so why is he playing? The proper response to such objections, from the point of view of someone who wishes to understand real-world games, is that theories like Li s and Straffin s are prototypes of more sophisticated theories, not yet developed, that take into account the fact that players of real-life games are partly rational and partly emotional creatures, capable of such things as stupidity and duplicity It would be good to have a framework into which the theories of Li and Straffin, along with three-player game-theories of the future, can be fitted This neutral framework would make no special assumptions about how the players behave Here, we develop such a theory It is a theory designed to answer the single question Can I win?, asked by a single player playing against two adversaries of unknown characteristics Not surprisingly, the typical answer given by the theory is No ; in most positions, any two players can gang up on the third But it turns out that there is a great deal to be said about those games in which one of the players does have a winning 1

strategy In addition to the coarse classification of three-player games according to who (if anyone) has the winning strategy, one can also carry out a fine classification of games analogous to, but much messier than, the classification of two-player games according to Grundy-value The beginnings of such a classification permit one to answer the riddle with which this article opened; the later stages lead to many interesting complications which have so far resisted all attempts at comprehensive analysis 1 NOTATION AND PRELIMINARIES Games will be denoted by the capital letters,,, and As in the two-player theory, we can assume that every position carries along with it the rules of play to be applied, so that each game may be identified with its initial position The game is an option of if it is legal to move from to To build up all the finite games, we start from the nullgame (the unique game with no options) and recursively define as the game with options The game will be denoted by 1, the game will be denoted by 2, and so on (It should always be clear from context whether a given numeral denotes a number or a Nim game) We recursively define the relation of identity by the rule that and are identical if and only if for every option of there exists an option of identical to it, and vice versa We define (disjunctive) addition, represented by, by the rule that is the game whose options are precisely the games of the form and It is easy to show that identity is an equivalence relation that respects the bracketing and addition operations, that addition is associative and commutative, and that is an additive identity The following abbreviations will prove convenient: Thus, means means ( times) means ( layers deep) denotes (We ll never need to talk about Nim-heaps of size, so our juxtaposition convention won t cause trouble) Note that for all, the games,,, and are identical Relative to any non-initial position in the course of play, one of the players has just moved (the Previous player) and one is about to move (the Next player); the remaining player is the Other player At the start of the game, players Next, Other, and Previous correspond to the first, second, and third players (even though, strictly speaking, there was no previous move) We call a Next-game ( -game) if there is a winning strategy for Next, and we let be the set of -games; is the type of, and belongs to We define -games and -games in a similar way If none of the players has a winning strategy, we say that is a Queer game ( -game) In a slight abuse of notation, I will often use to mean belongs to, and use the letters to stand for unknown games belonging to these respective types Thus I will write,,, etc; and the problem posed in the Introduction can be formulated succinctly as: solve or prove that no solution exists (At this point I invite the reader to tackle There is a simple and elegant solution) The following four rules provide a recursive method for classifying a game: (1) is an -game exactly if it has some -game as an option (2) is an -game exactly if all of its options are - games, and it has at least one option (this proviso prevents us from mistakenly classifying as an -game) (3) is a -game exactly if all of its options are -games (4) is a -game exactly if none of the above conditions is satisfied Using these rules, it is possible to analyze a game completely by classifying all the positions in its game-tree, from leaves to root 2 SOME SAMPLE GAMES Let us first establish the types of the simpler Nim games It s easy to see that and so on; in general, the type of is,, or according as the residue of mod 3 is 0, 1, or 2 Also and so on, because in each case Next can win by taking the whole heap and so on; in general, the type of is,, or according as the residue of mod 3 is 0, 1, or 2 The winning strategy for these -games is simple: reduce the game to one of the -positions is a solution of the equation Does imply that in general? We can easily see that the answer is No : 2

( is identical to, so they can be treated as a single option) Here are some more calculations which will be useful later Table 3 shows the twenty-two satisfiable equations and their solutions Equation Solution Equation Solution 3 ADDING GAMES The type of is not in general determined by the types of and (For example, 1 and 2 are both of type, but while ) That is, addition does not respect the relation belongs to the same type as To remedy this situation we define equivalence ( ) by the condition that if and only if for all games, and belong to the same type It is easy to show that equivalence is an equivalence relation, that it respects bracketing and addition, and that if then (that is, equivalence options of a game may be conflated) We are now in a position to undertake the main task of this section: determining the addition table Recall that in the twoplayer theory, there are only two types ( and ) and their addition table is as shown in Table 1 Table 1 The two-player addition table Here, the entry denotes the fact that the sum of two - games can be either a -game or an -game The analogous addition table for three-player games is given by Table 2 Table 2 The three-player addition table Notice that in one particular case (namely and, or vice versa), knowing the types of and does tell one which type belongs to, namely A corollary of this is that To prove that Table 1 applies, one simply finds solutions of the allowed equations, (from which follows),, and, and proves that the forbidden equations and have no solutions To demonstrate the validity of Table 2, we must find solutions to twenty-two such equations, and prove that the remaining eighteen have no solutions Table 3 Some sums And now, the proofs of impossibility for the eighteen impossible cases Claim 1 None of the following is possible (1) (2) (3) (4) (5) Proof By (joint) infinite descent Here, as in subsequent proofs, the infinite-descent boilerplating is omitted Note that none of the hypothetical -games in equations (1)-(4) can be the -game, so all of these games have options Suppose (1) holds; say,, Some option or must be a -game But then we have either (every option must be an - game), which is (2), or (every option must be an -game), which is (3) Suppose (2) holds; say,, Then there exists, which must satisfy (equation (4)) Suppose (3) holds; say,, Then there exists, which must satisfy (equation (5)) Suppose (4) holds; say,, Then there exists, which must satisfy (equation (1)) Finally, suppose (5) holds; say,, Then there exists, which must satisfy (equation (1)) Claim 2 None of the following is possible (6) (7) (8) 3

Proof By infinite descent A solution to (6) yields an (earliercreated) solution to (7), which yields a solution to (8), which yields a solution to (6) Claim 3 It is impossible that Proof By contradiction A solution to (9) would yield a solution to (8) Claim 4 None of the following is possible (9) (10) (11) (12) (13) (14) Proof By infinite descent (making use of earlier results as well) Suppose (10) holds with Some option or must be a -game In the former event, we have (since ), so that either (equation (2)), (equation (7)), or (equation (11)); in the latter event we have (equation (12)) Suppose (11) holds with Since, it has an option of type or type (for if all options of were - games and -games, would be of type or ) If, then we have (equation (8)), and if, then we have (equation (13)) Suppose (12) holds with Then (equation (14)) Suppose (13) holds with Since, it has an option of type or of type (for if all options of were - games and -games, would be of type or ) yields (equation (1)), and yields (equation (10)) Finally, suppose (14) holds with Then there exists, which must satisfy (equation (10)) Claim 5 It is impossible that (15) Proof By contradiction A solution to (15) would yield a solution to (13) Claim 6 Neither of the following is possible: (16) (17) Proof By infinite descent Suppose (16) holds with Then some option of must be a -game; without loss of generality, we assume But, and we have already ruled out (equation (11)), (equation (5)), and (equation (12)), so we have (equation (17)) Suppose (17) holds with must have an -option or -option, but if then (equation (14)), which can t happen; so Similarly, has a -option, so (equation (16)) (Note that the second half of this proof requires us to look two moves ahead, rather than just one move ahead as in the preceding proofs) The remaining case is surprisingly hard to dispose of; the proof requires us to look five moves ahead Claim 7 It is impossible that (18) Proof By infinite descent Suppose (18) holds with For all we have, so that must have some -option; but this -option cannot be of the form, since (equation (9)) Hence there must exist an option of such that This implies that, since none of the cases (equation (3)), (equation (7)), (equation (12)) can occur Similarly, every has an option such that, Since is a -game, and are -games and is an -game One of the options of must be a -game; without loss of generality, say Since and since none of the cases (equation (9)), (equation (2)), (equation (15)) can occur, must be an -game But recall that is a -game, so that its option is an -game This gives us, which is an earlier-created solution to (18) The proof of Claim 7 completes the proof of the validity of Table 2 Observe that this final clinching claim, which answers the article s opening riddle in the negative, depends on five of the preceding six claims Our straightforward question thus seems to lack a straightforward solution In particular, one would like to know of a winning strategy for the Natalieand-Percival coalition in the game that makes use of Oliver s winning strategies for and Indeed, it would be desirable to have strategic ways of understanding all the facts in this section At this point it is a good idea to switch to a notation that is more mnemonically helpful than,, and, vis-à-vis addition Let,, and denote the Nim-positions,,, respectively Also, let be the Nim-position (Actually, we ll want these symbols to represent the equivalence classes of these respective games, but that distinction is unimportant right now) We will say that two games, are similar if they have the same type; in symbols, Every game is thus similar to exactly one of,,, and We can thus use these four symbols to classify our games by type; for instance, instead of writing, we can write 4

Here is the rule for recursively determining the type of a game in terms of the types of its options, restated in the new notation: (1) is of type exactly if it has some option of type (2) is of type exactly if all of its options are of type, and it has at least one option (3) is of type exactly if all of its options are of type game theory, a strategy-stealing argument can be used to show that the sum of a game of type with itself must be of type (even though a sum of two distinct games of type can be of either type or type ) We seek a similar understanding of what happens when we add a three-player game to itself Table 6 shows the possible types can have in our three-player theory, given the type of (4) is of type exactly if none of the above conditions is satisfied Here is the new addition table for 3-player game types; it resembles a faulty version of the modulo 3 addition table Table 6 The doubling table Table 4 The new, improved three-player addition table It is also worthwhile to present the subtraction table as an object of study in its own right To this end define as an alternative to none All All All All All Table 5 The three-player subtraction table The minuend is indicated by the row and the subtrahend by the column Note that subtraction is not a true operation on games; rather, the assertion is means that if are games such that and then,, or The six entries in the upper left corner of the subtraction table (the only entries that are single types) correspond to assertions that can be proved by joint induction without any reference to earlier tables In fact, a good alternative way to prove that addition satisfies Table 4 would be to prove that addition satisfies the properties implied by the six upper-left entries in Table 5 (by joint induction) and then to prove three extra claims: (i) if and then ; (ii) if and then ; and (iii) if and then 4 ADDING GAMES TO THEMSELVES Another sort of question related to addition concerns the disjunctive sum of a game with itself Recall that in two-player To verify that all the possibilities listed here can occur, one can simply look at the examples given at the beginning of Section 3 To verify that none of the omitted possibilities can occur, it almost suffices to consult Table 4 The only possibility that is not ruled out by the addition table is that there might be a game with, Suppose were such a game Then would have to have a -option (now we call it a -option) along with another option such that This implies that and Since, the condition implies (by way of Table 4) that But implies (by way of Table 4) that or This contradiction shows that no such game exists, and completes the verification of Table 6 In the same spirit, we present a trebling table (Table 7), showing the possible types can have given the type of Table 7 The trebling table To prove that all the possibilities listed in the first three rows can actually occur, one need only check that,,,,, and To prove that the nine cases not listed cannot occur takes more work Four of the cases are eliminated by the observation that can never be of type (the second and third players can always make the Next player lose by using a copy-cat strategy) Tables 3 and 5 allow one to eliminate three more cases The next two claims take care of the final two cases Claim 8 If, then Proof Suppose with Let be an option of Since, must have 5

a -option of the form (for some option of ) or of the form (for some option of ) In either case, we find that the -game, when added to some other game ( or ), yields a game of type ; this is impossible, by Table 4 We start our proof of the validity of Table 8 by showing that no two games in the table are equivalent to each other In this we will be assisted by Tables 9 and 10 Claim 9 If, then Proof Suppose with Notice that for every option of Case I: There exist options, of (possibly the same option) for which Then its option Since, Table 5 gives But this contradicts Table 6, since Case II: There do not exist two such options of Let be an option of Since, and since there exists no for which, there must exist an option of such that or, by Table 6, but cannot be of type, since adding yields a -position Hence, and Table 5 implies Since, there must exist an option with Everything we ve proved so far about applies equally well to (since all we assumed about was that it be some option of ) In particular, must have an option such that However, since is an option of the -position, Hence and are two -positions whose sum is a - position, contradicting Table 4 Table 9 The type of Table 9 gives the types for games of the form Each row of the chart gives what we shall call the signature of, relative to the sequence Since no two games of the form have the same signature, no two are equivalent Similarly, Table 10 is the signature table for games of the form, relative to 5 NIM FOR THREE We wish to classify all Nim-positions as belonging to,,, or or rather, as we now put it, as being similar to,,, or We will actually do more, and determine the equivalence classes of Nim games Table 8 shows the games we have classified so far (on the left) and their respective types (on the right) Table 8 Basic positions of Nim We will soon see that every Nim-game is equivalent to one of the Nim-games in Table 8 We call these reduced Nimpositions The last paragraph of this section gives a procedure for converting a three-player Nim-position into its reduced form Throughout this section (and the rest of this article), the reader should keep in mind the difference between the notations 2 and The former is a single Nim-heap of size 2; the latter is the game-type that corresponds to a second-player win Note in particular that 2 is not of type but rather of type Table 10 The type of We see that all the games and are distinct What about? It can t be equivalent to for any (even though both are -games), because while What about 3? It can t be equivalent to for any, because while ; it can t be equivalent to because while ; it can t be equivalent to because while ; and it can t be equivalent to because while Now that we know that all of the Nim games in Table 8 are inequivalent, let us show that every Nim game is equivalent to one of these Claim 10 for all Proof Any two players can gang up on the third, by depleting neither heap until the victim has made his move, and then removing both heaps Claim 11 The following are true for all games : (a) for 6

(b) for (c) If then, for (d) for (e) for (f) for (g) for Proof (a) Suppose Then its options and are - games But since is also an option of, this is a contradiction (b) Suppose Then,, and are all -games, and in particular must have a -option That -option can be neither nor, so there must exist, contradicting (a) (c) Assume Then either or (no other option of can be of type, by (a)), and in either case (d) Suppose Then,, and are all - games must have a -option, but and no option or ( ) can be a -game (by (a)), so itself must be a -game Also, since and, there must exist with Then and, which is inconsistent with (e) Every option of has a component heap of size 2 or more, so has no -options, by (a) (f) Suppose Then can t be (by Claim 10), so it must have an option ;, contradicting (e) (g) Suppose Then can t be (by Claim 10), so it must have an option ;, contradicting (f) Note that (e), (f), and (g) together imply that for all Claim 12 The following are true for all games : (A) for (B) for (C) for Proof (A) Take an arbitrary game We know that each of, is either of type or type (by (a), (b) above) If either of them is a -game, then so is the other (by (c)), and if neither of them is a -game, then both are -games Either way, and have the same type (B) The proof is similar, except that one needs (d) instead of (b) (C) For all, and To reduce a given Nim-position to one of the previously tabulated forms, first replace every by 3 This puts in the form If, then we have Otherwise, we have in the form,, or Since, the last of these cases can be reduced to unless 6 EQUIVALENCE CLASSES The Nim game has the property that if one adds to it any other Nim-position, one gets a game of type In fact, if one adds any game whatsoever to, one still gets a game of type is thus an element of an important equivalence class, consisting of all games such that for all games We call this class the equivalence class of infinity This equivalence class is a sort of a black hole, metaphorically speaking; add any game to the black hole, and all you get is the black hole If you take a two-player game for which a nice theory exists and study the three-player version, then it is unfortunately nearly always the case that most of the positions in the game are in the equivalence class of infinity There are some games which are close to infinity Paradoxically, such games can give us interesting information about games that are very far away from infinity Consider, for instance, the -game (the game whose sole option is a Nim-heap of size 2) Claim 13 The only game for which is the game 0 Proof Let be the simplest game not identical to 0 such that Case I: Then But Claim 11(b), together with the fact that is equivalent to every Nim-position with, tells us that this can t happen Case II: The winning option of can t be, by Claim 11(a), so it must be an option of the form But then, which contradicts the assumed minimality of ( won t help us, since, not ) Case III: Letting be any option of, we have This contradicts the assumed minimality of This implies that no game is equivalent to 0 7 OPEN QUESTIONS Question 1 How do the doubling and tripling tables (Tables 6 and 7) extend to higher compound sums of a game with itself? Question 2 Is there a decision procedure for determining when two impartial three-player games are equivalent to each other? Question 3 What does the neighborhood of infinity look like? The game has the property that when you add it to any non-trivial game, you get Is there a game of type with this property? Is there one of type with this property? Question 4 How does the theory generalize to players, with? It is not hard to show that the portion of Table 5 in the upper left corner generalizes to the case of more than three players in a straightforward way However, carrying the 7

theory beyond this point seems like a large job Here are two particular questions that seem especially interesting: Can an -fold sum of a game with itself be a win for any of the players other than the th? Does there exist a black hole such that for all games, is a win for any coalition with over half the players? 8 ACKNOWLEDGMENTS This research was supported by a Knox Fellowship from Harvard College I express deep appreciation to John Conway for his encouragement and for stimulating conversations I also thank Richard Guy and Phil Straffin for many helpful remarks on the manuscript [1] E R Berlekamp, J H Conway, and R K Guy, Winning Ways, for your mathematical plays, Academic Press, 1982 [2] J H Conway, On Numbers and Games, Academic Press, 1976 [3] S-Y R Li, -person Nim and -person Moore s games, Internat J Game Theory 7 (1978), 31-36 [4] P D Straffin, Jr, Three-person winner-take-all games with Mc- Carthy s revenge rule, College J Math 16 (1985), 386-394 8